New subclasses of the class of close-to-convex functions

Abstract:

In this paper we introduce new subclasses of the class of close-to-convex functions. We call a regular function $f(z)$ an alpha-close-to-convex function if $(f(z)f’(z)/z) \ne 0$ for z in E and if for some nonnegative real number $\alpha$ there exists a starlike function $\phi (z) = z + \cdots$ such that \[ \operatorname {Re} \;\left [ {(1 - \alpha )\frac {{zf’(z)}}{{\phi (z)}} + \alpha \frac {{(zf’(z))’}}{{\phi ’(z)}}} \right ] > 0\] for z in E. We have proved that all alpha-close-to-convex functions are close-to-convex and have obtained a few coefficient inequalities for $\alpha$-close-to-convex functions and an integral formula for constructing these functions. Let ${\mathfrak {F}_\alpha }$ be the class of regular and normalised functions $f(z)$ which satisfy $\operatorname {Re} \;(f’(z) + \alpha zf''(z)) > 0$ for z in E. $f(z) \in {\mathfrak {F}_\alpha }$ gives $\operatorname {Re} f’(z) > 0$ for z in E provided $\operatorname {Re} \alpha \geqslant 0$. A sharp radius of univalence of the class of functions $f(z)$ for which $zf’(z) \in {\mathfrak {F}_\alpha }$ has also been obtained.